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Modelling Extreme Rainfall in the Era of Climate Change Concerns: Towards a Consistent Stochastic Methodology

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Modelling Extreme Rainfall in the Era of Climate Change Concerns: Towards a Consistent Stochastic Methodology School for Young Scientists: “Modelling and forecasting of river flows and managing hydrological risks: towards a new generation of methods” Moscow, Russia, 22-26 October 2018 Modelling extreme rainfall in the era of climate change concerns: Towards a consistent stochastic methodology One Step Forward, Two Steps Back Demetris Koutsoyiannis Department of Water Resources and Environmental Engineering School of Civil Engineering National Technical University of Athens, Greece ([email protected], http://www.itia.ntua.gr/dk/) Presentation available online: http://www.itia.ntua.gr/1897/ Greetings from the Itia research team http://www.itia.ntua.gr/ D. Koutsoyiannis, Modelling extreme rainfall 1 Giants of the Moscow School of Mathematics D. Koutsoyiannis, Modelling extreme rainfall 2 On names and definitions: A contrast Ἀρχὴ παιδεύσεως ἡ τῶν ὀνομάτων ἐπίσκεψις” What’s in a name? That which (The beginning of education is the inspection of we call a rose, by any other names) name would smell as sweet. Attributed to Socrates by Epictetus, Discourses, Ι.17,12 William Shakespeare, “Romeo and Juliet” (Act 2, scene 2) Each definition is a piece of secret ripped from Let me argue that this Nature by the human spirit. I insist on this: any situation [absence of a complicated thing, being illumined by definitions, definition] ought not create being laid out in them, being broken up into concern and steal time from pieces, will be separated Into pieces completely useful work. Entire fields of transparent even to a child, excluding foggy and mathematics thrive for dark parts that our intuition whispers to us while centuries with a clear but acting, separating into logical pieces, then only evolving self-image, and can we move further, towards new successes due nothing resembling a to definitions . definition Nikolai Luzin (from Graham and Kantor, 2009) Benoit Mandelbrot (1999, p. 14) D. Koutsoyiannis, Modelling extreme rainfall 3 Part A Premises for a stochastic framework about change D. Koutsoyiannis, Modelling extreme rainfall 4 The general framework: Seeking theoretical consistency in analysis of geophysical data (Using stochastics) D. Koutsoyiannis, Modelling extreme rainfall 5 Why stochastics in geophysics and hydrology? Geophysics is the branch of physics that relies most decisively on data. Geophysical data are numbers but to treat them we need to use stochastics, not arithmetic. Stochastics is the mathematics of random variables and stochastic processes. Random variables and stochastic processes are abstract mathematical objects, whose properties distinguish them from typical variables that take on numerical values. It is important to understand these properties before making calculations with data, otherwise the results may be meaningless (not even wrong). The numerical data allow us to estimate (not to determine precisely) expectations. Expectations are defined as integrals of products of functions. For a continuous random variable x with probability density function f(x), the expectation of an arbitrary function g of x (where g(x) ∞ ( ) ( ) . is a random variable per se), the expectation of g(x) is defined as 휃 ≔ E[푔(푥)] ≔ ∫−∞ 푔 푥 푓 푥 d푥 Central among expectations are the moments, in which g(x) is a power of x (or a linear expression of x). To estimate true parameters 휃 from data we need estimators; the estimator 휃̂ of θ is a random variable depending on the stochastic process of interest x(t) and is a model per se, not a number. The estimate 휃̂ is a number, calculated by using the observations and the estimator. Characteristic statistics of the estimator 휃̂ are its bias, E[휃̂] − 휃, and its variance var[휃̂]. When E[휃̂] = 휃 the estimator is called unbiased. Estimation is made possible thanks to two concepts of stochastics: stationarity and ergodicity. Stationarity and ergodicity are not incompatible with, or contradictory to, change. D. Koutsoyiannis, Modelling extreme rainfall 6 Change is crucial in Hydrology: ‘Panta Rhei’—The scientific decade of IAHS 2013-2022 http://iahs.info/Commissions--W-Groups/Working-Groups/Panta-Rhei.do D. Koutsoyiannis, Modelling extreme rainfall 7 ‘Panta Rhei’: © Heraclitus Change and randomness Πάντα ῥεῖ Everything flows (Heraclitus; quoted in Plato’s Cratylus, 339-340) Αἰών παῖς ἐστι παίζων πεσσεύων Time is a child playing, throwing dice (Heraclitus; Fragment 52) D. Koutsoyiannis, Modelling extreme rainfall 8 Change, logic, precision: © Aristotle Μεταβάλλει τῷ χρόνῳ πάντα All is changing in the course of time ( Aristotle; Meteorologica, I.14, 353a 16) Λογική, συλλογισμός, επαγωγή, ορθός λόγος Logic, deduction, induction, (right) reason (Aristotle, Organon & Nicomachean Ethics) …τοσοῦτον τἀκριβὲς ἐπιζητεῖν καθ᾽ ἕκαστον γένος, ἐφ᾽ ὅσον ἡ τοῦ πράγματος φύσις ἐπιδέχεται … look for precision in each class of things just so far as the nature of the subject admits (Aristotle, Nicomachean Ethics 1094b) D. Koutsoyiannis, Modelling extreme rainfall 9 Hurst-Kolmogorov dynamics—Or: Earth’s perpetual change 7 Annual Nile River annual minimum Structured 6 30-year average water level (849 values) random 5 4 3 Minimum water depth (m) 2 1 0 600 700 800 900 1000 1100 1200 1300 1400 1500 Year AD Nilometer data: Koutsoyiannis (2013a) 7 "Annual" Each value is the minimum of m=36 roulette wheel Purely 6 30-"year" average outcomes. The value of m was chosen so that the standard deviation be equal to the Nilometer series random 5 4 3 2 1 Minimum roulette wheel outcome 0 600 700 800 900 1000 1100 1200 1300 1400 1500 "Year" D. Koutsoyiannis, Modelling extreme rainfall 10 The climacogram: A simple statistical tool to quantify change across time scales • Take the Nilometer time series, x1, x2, ..., x849, and calculate the sample estimate of variance γ(1), where the superscript (1) indicates time scale (1 year) • Form a time series at time scale 2 (years): (2) 푥1 + 푥2 (2) 푥3 + 푥4 (2) 푥847 + 푥848 푥 ≔ , 푥 ∶= , . , 푥 ∶= (1) 1 2 2 2 424 2 and calculate the sample estimate of the variance γ(2). • Repeat the same procedure and form a time series at time scale 3, 4, … (years), up to scale 84 (1/10 of the record length) and calculate the variances γ(3), γ(4),… γ(84). • The climacogram is the variance γ (κ) as a function of scale κ; it is visualized as a double logarithmic plot of γ (κ) vs. κ (or alternatively of the standard deviation σ(κ)). • If the time series xτ represented a pure random process, the climacogram would be a straight line with slope –1 (the proof is very easy). • In real world processes, the slope is different from –1, designated as 2H – 2, where H is the so-called Hurst coefficient (0 < H < 1). • The scaling law γ(κ) = γ(1) / κ2 – 2H defines the Hurst-Kolmogorov (HK) process. • High values of H (> 0.5) indicate enhanced change at large scales, else known as long-term persistence, or strong clustering (grouping) of similar values. D. Koutsoyiannis, Modelling extreme rainfall 11 The climacogram of the Nilometer time series 7 • The Hurst-Kolmogorov process Annual 6 30-year average seems consistent with reality. 5 • The Hurst coefficient is H = 0.87 4 (Similar H values are estimated 3 from the simultaneous record of Minimum water depth (m) 2 maximum water levels and from 1 0 the modern, 131-year, flow 600 700 800 900 1000 1100 1200 1300 1400 1500 record of the Nile flows at Year AD Aswan). 1 • The Hurst-Kolmogorov behaviour, seen in the climacogram, indicates that: Variance (m²) (a) long-term changes are more frequent and intense 0.1 than commonly perceived, and (b) future states are much Empirical (from data) more uncertain and Purely random (H = 0.5) Markov unpredictable on long time Hurst-Kolmogorov, theoretical (H = 0.87) Hurst-Kolmogorov adapted for bias horizons than implied by pure 0.01 randomness. 1 10 100 Scale (years) D. Koutsoyiannis, Modelling extreme rainfall 12 Change and predictability Change Koutsoyiannis, 2013a Predictable Unpredictable (regular) (random) Non-periodic Periodic Purely random Structured e.g. acceleration of e.g. daily and e.g. consecutive random outcomes of dice e.g. climatic a falling body annual cycles fluctuations Simple systems – Short time horizons Complex systems – Long time horizons Important but trivial Most interesting D. Koutsoyiannis, Modelling extreme rainfall 13 The cause of change: © Peter Atkins Atkins, 2004 Atkins, 2007 D. Koutsoyiannis, Modelling extreme rainfall 14 Entropy ≡ Uncertainty quantified • Historically entropy was introduced in thermodynamics but later it was given a rigorous definition within probability theory (owing to Boltzmann, Gibbs and Shannon). • Thermodynamic and probabilistic entropy are essentially the same thing (Koutsoyiannis, 2010, 2013b, 2014; but others have different opinion). • Entropy acquires its importance from the principle of maximum entropy (Jaynes, 1957), which postulates that the entropy of a random variable should be at maximum, under the conditions (constraints) which incorporate the available information about this variable. • The tendency of entropy to become maximal explains a spectrum of phenomena from the random outcomes of dice to the Second Law of thermodynamics as the driving force of natural change. • Entropy is a dimensionless measure of uncertainty: Discrete random variable z Continuous random variable z 푓(z) ∞ 푓(푧) Φ[z] := E[–ln P(z)] = ∑푤 푃 ln 푃 Φ[z] := E[– ln ] = – ∫ ln 푓(푧)d푧 푗 = 1 푗 푗 ℎ(z) −∞ ℎ(푧) where P ≔ P{z = z } (probability) where f(z) is probability density and h(z) j j is the density of a background measure D. Koutsoyiannis, Modelling extreme rainfall 15 Memorable moments in the history of stochastics Ludwig Boltzmann George D. Birkhoff Aleksandr Khinchin Andrey N. Kolmogorov (1844 –1906, Universities of (1884 – 1944; (1894 – 1959; Moscow (1903 – 1987; Moscow State Graz and Vienna, Austria, and Princeton, Harvard, State University, University, Russia) Munich, Germany) USA) Russia) 1877 Explanation of the 1931 Discovery of 1933 Purely measure- 1931 Introduction of the terms concept of entropy in the ergodic theoretic proof of the process to describe change of a probability theoretic context.
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